Let $f : P\to P$ be a bijective map from a plane $P$ to itself such that: (i) $f (r)$ is a line for every line $r$, (ii) $f (r) $ is parallel to $r$ for every line $r$. What possible transformations can $f$ be?

Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc. (a) Prove the relation \[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \] where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that \[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\] (b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function with the intermediate value property. If $f$ is injective on $\mathbb{R} \setminus \mathbb{Q},$ prove that $f$ is continuous on $\mathbb{R}.$ [i]Julieta R. Vergulescu[/i]

We call [i]word [/i] a sequence of letters $\overline {l_1l_2...l_n}, n\ge 1$ . A [i]word [/i] $\overline {l_1l_2...l_n}, n\ge 1$ is called [i]palindrome [/i] if $l_k=l_{n-k+1}$ , for any $k, 1 \le k \le n$. Consider a [i]word [/i] $X=\overline {l_1l_2...l_{2014}}$ in which $ l_k\in\{A,B\}$ , for any $k, 1\le k \le 2014$. Prove that there are at least $806$ [i]palindrome [/i] [i]words [/i] to ''stick" together to get word $X$.

Let $G$ be a group and $A$ a nonempty subset of $G$. Let $AA=\{xy\mid x,y\in A\}$. [list=a] [*] Prove that if $G$ is finite, then $AA=A$ if and only if $|A|=|AA|$ and $e\in A$. [*] Give an example of a group $G$ and a nonempty subset $A$ of $G$ such that $AA\neq A$, $|AA|=|A|$ and $AA$ is a proper subgroup of $G$. [/list] [i]Mathematical Gazette - Robert Rogozsan[/i]

Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

Given positive integer $M.$ For any $n\in\mathbb N_+,$ let $h(n)$ be the number of elements in $[n]$ that are coprime to $M.$ Define $\beta :=\frac {h(M)}M.$ Proof: there are at least $\frac M3$ elements $n$ in $[M],$ satisfy $$\left| h(n)-\beta n\right|\le\sqrt{\beta\cdot 2^{\omega(M)-3}}+1.$$ Here $[n]:=\{1,2,\ldots ,n\}$ for all positive integer $n.$ [i]Proposed by Bin Wang[/i]

Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.

Let $p_1$, $p_2$, $...$, $p_n$ be different prime numbers ($n\ge 2$). All possible products containing an even number of coefficients (all coefficients are different) are composed of these numbers. Let $S_n$ be the sum of all such products. For example, $$S_4 = p_1p_2 + p_1p_3 + p_1p_4 + p_2p_3 + p_2p_4 + p_3p_4+ p_1p_2p_3p_4.$$ Prove that $S_n + 1$ is divisible by $2^{n-2}$.

Let $ABCD$ be a cyclic quadrilateral such that the lines $AB$ and $CD$ intersects in $K$, let $M$ and $N$ be the midpoints of $AC$ and $CK$ respectively. Find the possible value(s) of $\angle ADC$ if the quadrilateral $MBND$ is cyclic.

Alice and Bob each flip $20$ fair coins. Given that Alice flipped at least as many heads as Bob, what is the expected number of heads that Alice flipped? [i]Proposed by Adam Bertelli[/i]

$P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$, $Q_3$ are distinct points in the plane. The distances $P_1Q_1$, $P_2Q_2$, $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$, and $P_2P_3$ and $Q_3Q_2$. Show that $P_1Q_1$, $P_2Q_2$ and $P_3Q_3$ intersect in a point.

$ABC$ is an acute-angled triangle. Let $D$ be the point on $BC$ such that $AD$ is the bisector of $\angle A$. Let $E, F$ be the feet of perpendiculars from $D$ to $AC,AB$ respectively. Suppose the lines $BE$ and $CF$ meet at $H$. The circumcircle of triangle $AFH$ meets $BE$ at $G$ (apart from $H$). Prove that the triangle constructed from $BG$, $GE$ and $BF$ is right-angled.

Let $ABC$ be a triangle and let $H$ be the orthogonal projection of $A$ on the line $BC$. Let $K$ be a point on the segment $AH$ such that $AH = 3 KH$. Let $O$ be the circumcenter of triangle $ABC$ and let $M$ and $N$ be the midpoints of sides $AC$ and $AB$ respectively. The lines $KO$ and $MN$ meet at a point $Z$ and the perpendicular at $Z$ to $OK$ meets lines $AB, AC$ at $X$ and $Y$ respectively. Show that $\angle XKY = \angle CKB$. [i]Italy[/i]

The following is known about the reals $ \alpha$ and $ \beta$ $ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$ Determine $ \alpha+\beta$

[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality $$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$ holds. [b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?

Let $ABC$ be a triangle, let $A_1, B_1, C_1$ be the antipodes of the vertices $A, B, C$, respectively, in the circle $ABC$, and let $X$ be a point in the plane $ABC$, collinear with no two vertices of the triangle $ABC$. The line through $B$, perpendicular to the line $XB$, and the line through $C$, perpendicular to the line $XC$, meet at $A_2$, the points $B_2$ and $C_2$ are defined similarly. Show that the lines $A_1A_2, B_1B_2$ and $C_1C_2$ are concurrent.

Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

Let $a$ be the sum of the numbers: $99 \times 0.9$ $999 \times 0.9$ $9999 \times 0.9$ $\vdots$ $999\cdots 9 \times 0.9$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$. Find the sum of the digits in the number $a$.

Given are $5$ segments, such that from any three of them one can form a triangle. Prove that from some three of them one can form an acute-angled triangle.

Sequence $(a_n)$ satisfies that $3a_{n+1}+a_n=4(n\geq1),a_1=9$, let $S_n=\sum_{i=1}^{n}a_i$, then the minumum value of $n$ such that $|S_n-n-6|<\frac{1}{125}$ is $\text{(A)}5\qquad\text{(B)}6\qquad\text{(C)}7\qquad\text{(D)}8$

For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

In a sphere $S_0$ we radius $r$ a cube $K_0$ has been inscribed. Then in the cube $K_0$ another sphere $S_1$ has been inscribed and so on to infinity. Calculate the volume of all spheres created in this way.

In a group of five people any two are either friends or enemies , no three of them are friends of each other and no three of them are enemies of each other . Prove that every person in this group has exactly two friends .